INTRODOCTION TO

EUCLID’S

GEOMETRY

Introduction To Geometry Geometry Is a branch of mathematics concerned

with questions of shape , size , relative position of figures, and the properties of space. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.

Necessity Of Geometry Geometry is everywhere.  Angles, shapes, lines,

line segments, curves, and other aspects of geometry are every single place you look, even on this page.  Letters themselves are constructed of lines, line segments, and curves!  Take a minute and look around the room you are in, take note of the curves, angles, lines and other aspects which create your environment.  Notice that some are two-dimensional while others are three-dimensional.  These man-made geometrical aspects please us in an aesthetic way

Contribution of THALES ,

PYTHAGOREAS & EUCLID

Thales He was born around 624 BC and died around 547 BC. Yes that was a long time ago, but he made some very major contributions to the field of geometry. In fact, some consider him the first mathematician. On a visit to Egypt, he was able to calculate the height of a pyramid. He is credited for making five notable contributions to the field of geometry, one of which is named after him.

The first is that the diameter of a circle bisects, or cuts, the circle in half. The second is that the base angles of an isosceles triangle are equal to each other. The third is when you have two straight lines intersecting each other, the opposite or vertical angles are equal to each other. The fourth notable contribution states that when two triangles have two equal angles and one equal side, then they are congruent, or equal, to each other. The fifth is called Thales' Theorem. It states that an angle that is inscribed or drawn inside a half-circle or semicircle will be a right angle. These five contributions are credited to Thales because he provided the first written proof of these theorems.

Pythagoreas Pythagoras was born in approximately 569 B.C. His father was Mnesarchus

and his mother was Pythais. Pythagoras spent his early years in Samos. There is little known about his child hood and all physical descriptions of Pythagoras are said to be fictitious except for the vivid birthmark on his thigh. It is believed that he had two brothers and some believe there were three. Pythagoras was extremely well educated. There were three philosophers that influenced him while he was young. One of the most important of these man was a man named Pherekydes. The philosopher that introduced Pythagoras to mathematical ideas was Thales , who lived in Miletus. It was because of Thales that Pythagoras became interested in math, astronomy and cosmology. Pythagoras was interested in all principles of mathematics. He was intrigued by the concept of numbers and basically numbers themselves. Pythagoras had a theory that all relations were able to be reduced in to some form of number. Pythagoras also derived a theory on ratios and scales being produced with the sound of vibrating strings. He made large contributions to music theory.He studied many different types of numbers ,for example triangles, odd numbers and perfect squares. He believed that each number had its own personality traits and were all different and unique. For example ten is the best number because it contains four consecutive integers (1+2+3+4=10)

Pythagoras’ greatest contribution to the mathematical society of today is Pythagoras theorem. It is believed that the theory of a2+b2=c2 was known to the Babylonians 1000 years before Pythagoras but it was he who was able to prove it.

The following are a list of theorems contributed by Pythagoras :-1.The sum of the angle is a triangles equal to two right angles. 2.The Pythagorean theorem. 3.Construction figures of a given area and geometrical algebra. 4.The discovery of irrationals 5.The five regular solids. 6. Pythagoras taught that the earth was a sphere in the center of the universe. Pythagoras life came to an end in approximately 475 B.C. Many of his contributions are still used in everyday math of today's generation.

Euclid Euclid was an ancient Greek mathematician who lived in the Greek

city of Alexandria in Egypt during the 3rd century BCE. Euclid is often referred to as the 'father of geometry' and his book Elements was used well into the 20th century as the standard textbook for teaching geometry.

The most famous work by Euclid is the 13-volume set called Elements. This collection is a combination of Euclid's own work and the first compilation of important mathematical formulas by other mathematicians into a single, organized format. Thus, it made mathematical learning much more accessible. Elements also contains a series of mathematical proofs, or explanations of equations that will always be true, which became the foundation for Western math.

Euclid's Elements contains several axioms, or foundational premises so evident they must be true, about geometry. These include such basic principles as when two non-parallel lines will meet, that opposite angles of an isosceles triangle are equal, and how to find the area of a right triangle. Elements also contains geometric interpretations of algebra, such as ideas like a(b+c)=ab+ac. Most important among these is Euclid's algorithm, a formula for devising the greatest common factor of two integers.

EUCLID’S DEFINITIONS AND AXIOMS

WITH EXPLANATIONS

AXIOMS EXPLANATIONS His third axiom would then be “if x=y, and if a=b, then x – a = y – b.” Euclid’s Fourth Axiom: Coincidental Equality The fourth axiom seems to be the most obvious reference to geometry. If two

shapes “coincide,” then one fills out the exact shape and volume of the second.

Simple cases include angles that are equal, straight line segments of the same length, and triangles of the same size and shape.

Consider drawing a triangle, and then constructing a second triangle in a way that copies the angles and lengths from the first triangle. Then, cut out the second triangle and lay it over the first. If these triangles precisely overlap, then they “coincide,” and are equal to one another.

Euclid’s Fifth Axiom: Part of the Whole Euclid’s fifth axiom states that “x + a > x.” To a modern mathematician, this

would not be true if ‘a’ had the value “zero,” or if ‘a’ were a negative number. For example, if ‘a’ were a geometric shape with no area, such as a line that has no thickness, then adding a line segment “beside” the edge of a square, ‘x’, would not increase the area of the square.

A more complete formula to cover our modern sensibilities would be “if a > zero, then x+a > x.”

Euclid’s Postulates Euclid’s First Postulate: a Line

Segment between Points Euclid’s first postulate states that any two points

can be joined by a straight line segment. It does not say that there is only one such line; it merely says that a straight line can be drawn between any two points.

Euclid’s Second Postulate: Extend a Straight Line

Euclid’s second postulate allows that line segment to be extended farther in that same direction, so that it can reach any required distance. This could result in an infinitely long line.

Euclid’s Third Postulate is Central to Circles

The third postulate starts with an arbitrary line segment, and an arbitrary point, which is not necessarily on the line segment. First, use the compass to note the end points of the line segment, then, put the sharp spike of the compass on the arbitrary point, and finally, draw the circle with the same radius as the line segment.

Euclid’s Fourth Postulate: All Right Angles are Equal to one another

Euclid was probably thinking of right angles as made by constructing one line perpendicular to another. Any two such right angles are “equal” to one another.

Euclid’s fourth postulate states that, “if x and y are both right angles, then x=y.”

This may be more profound if the angles are oriented differently: opening to the left or right, up or down, or towards some other direction.

Euclid did not measure angles in degrees or radians, and he did not use a protractor. Instead, he usually discusses “how many angles in a diagram add up to some number of right angles.” For example, in Book I, Proposition 13 basically states that when a straight line “stands on” another straight base line, the sum of the two angles on the base line adds up to two right angles.

Euclid’s Fifth Postulate: the Parallel Postulate

A parallelogram demonstrates parallel lines Euclid’s fifth postulate is the longest, and is now

called the “parallel postulate.” The fifth postulate has been the subject of much

debate and labour over the centuries -can it be proven from the other postulates and axioms? Eventually mathematicians realized that the fifth postulate defines plane geometry, the geometry for a flat surface, and it cannot be derived from the other Euclidean axioms.

This postulate’s explanation needs diagrams. Consider this paralellogram, with the interior angles on the base marked in green. The sum of those two interior angles is 180 degrees. The two blue slanted lines are parallel; they will never meet even if extended infinitely far.

Questions Based On Euclid’s

Geometry

Q1: If a point C lies between two points A and B such that AC

= BC, then prove that AC = AB/2.

Answer:AC = CB (Given)Also AC + AC = BC + AC. (Equals are added to

equals)∵ things which coincide with one another are

equal to one another. ∴ AC + BC coincides with AB⇒ 2AC = AB⇒ AC = ½AB.

Q2: Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question about the fifth

postulate.)

Answer: Euclid's postulate 5 states, "The whole is greater than the part." It is considered 'universal truth', because it holds true in every field.

Consider the following cases:Case I: Consider a group of numbers 15, 8, 4, 2, 1 such

that 15 = 8 + 4 + 2 + 1 and 15 is greater than any of its part (8, 4, 2, 1)

Case II: Consider a circle, consisting of six sectors (a, b, c, d, e and f).

The area of a circle as a whole is greater than that of any sector (its part).

Q3: How would you rewrite Euclid’s fifth postulate so that it would be

easier to understand?Answer:(i) For every line L and for every point P not

lying on L, there exists a unique line M passing through P and parallel to L.

If we draw perpendicular both from L and M i.e. AB and XY. The perpendicular distances are equal i.e. AB = XY.

(ii) Two distinct intersecting lines cannot be parallel to the same line.

Q & A from Exam. papers and other books

Q4: In Question 1, point C is called a mid-point of line segment AB. Prove that every line segment has

one and only ne mid-point.

Answer: Let there be two such mid points C and D. Then using above said theorem (see answer 4), we can prove

AC = ½AB ... (I)and AD = ½AB ... (II)From I and II, we have∴ AC = AD = ½ABAC and AD can be equal only if D coincides with C.

Therefore, C is the uniquemid-point.

MADE BY :- YASH – 43 YASH LAKRA – 45 ADITAY HOODA – 03 TUSHAR CHAUHAN – 40

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